Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system. Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that $T^{R(y)}(y)\in Y$.

Then we can define a function $F:Y\to Y$ by $F=T^R$ and consider the induced system $(Y,\mathcal{F}\cap Y, \mu|_Y,F)$.

Can we say that $F$ is ergodic?

When $R(x)=\inf\{n\ge 1 : T^n(x)\in Y\}$, this induced system is very well studied and the answer to my question is yes (see for example http://www.weizmann.ac.il/math/sarigo/sites/math.sarigo/files/uploads/ergodicnotes.pdf, Theorem 1.7). However, the given proof does not extend to the general return times I am considering above.

Thanks :)